You will find it easier to get around the Blue Mountains with a car.
I'd say car for the same reason Nick said.
For me it is unrestrictive as I prefer to go on my own schedule rather then a timetable of public transport.
There are parking charges near some of the major lookouts in the Blue Mountains so watch out for the signs. If you have the time it is well worth doing the "scenic circle" and going one way through Windsor, Richmond, Kurrajong and Bells Line of Rd. Far less traffic and very different views from the Great Western Highway.
On my previous trips (some time ago now) I took a train to Katoomba and then just rented a car for one day. You'd have to verify that there is still a rental agency in town. I found many of the local sights were quite walkable, so I only used the car to drive over to Jenolan Caves and sights on the way. However I would agree that a car will be needed to really appreciate the area. Just wanted to provide an option.
Alternatively, a short ride to Blackheath to the shops / cafes and a diversion to Govett's Leap and perhaps a short bush walk there is quite easy to do with a car.
And I also recommend Rob's suggestion of travelling down the mountain via Bells Line of Road - it's quite different to the drive up the HIghway.
Another lovely spot you may like is at Mt Tomah where the Botanic Gardens have an Annex - I'm not sure if there is still a charge to enter - it used to be $5 - as the entry fees have been lifted from Mt Annan Botanic Gardens.
FYI, here's a review of the Mt Tomah Botanic Gardens - tripadvisor.com.au/ShowUserReviews-g255058-d…
Re Vancouverite's suggestion of hiring a car in the Blue Mountains, see this thread from last year -
Since you are planning for June, I guess you are still planning so one more comment. If you choose to hire a car in Sydney to visit the Blue Mountains, may I suggest that you have a look at a map and perhaps consider the inland route North. There are many options. Head west through Bathurst, Orange to Dubbo and you come into some beautiful rolling hillsides. The Warrumbungle Ranges is majestic. Leaving Katoomba you could go to Lithgow, then Mudgee and work your way to the upper Hunter Valley, then you have the choice of the New England Highway or the Pacific Highway on the coast. Another option from Mudgee is to head towards Gunnedah - Google search the words of the poem "My Country" by Dorothea MacKellar - she lived in Gunnedah and it is a poem all Aussies (should) know! All the routes mentioned are sealed roads.
You could try a local car rental company in Katoomba - LTD Bus and Truck Rentals
We found it difficult to find information in the Blue Mountains, so we put together a couple of videos that are publicly available on youtube. One is on the town of Blackheath, where we stayed, and one is on the Jenolan Caves. It would be a shame to go to the Blue Mountains without going to the caves. We got a hire car, and found this useful, as there is no train line to the Jenolan Caves.
Here are the links:
Jenolan Caves: http://youtu.be/dbiZQjDhXl4
I hope this helps!!